Draw with , and . Draw the circumcircle of . Is the centre inside or outside the triangle?

Understand the Question To construct the triangle, draw base and construct angles and to locate vertex . The circumcentre of a triangle is the point of intersection of the perpendicular bisectors of its sides. The circumcircle is drawn with the circumcentre as the centre and the distance to any vertex as the radius. The position of the circumcentre depends on the angles of the triangle: Acute triangle: Circumcentre lies inside. Right triangle: Circumcentre lies on the hypotenuse (midpoint). Obtuse triangle: Circumcentre lies outside. Step 1 · Construct Triangle ABC 1. Draw a line segment . 2. At point , construct . 3. At point , construct . 4. Let the two rays intersect at point to form . Step 2 · Find Circumcentre O 1. Draw the perpendicular bisectors of sides , , and . 2. Mark their point of intersection as . Point is the circumcentre of . Step 3 · Draw Circumcircle 1. With as centre and radius equal to (or , ), draw a circle. 2. The circle passes through all three vertices , , and , forming the circumcircle. Step 4 · Determine Circumcentre Position Using the angle sum property of a triangle: Since all three angles (, , and ) are acute (), is an acute-angled triangle. Therefore, the circumcentre lies inside the triangle. Answer > The centre is inside the triangle. Common Mistakes Circumcentre vs Incentre: Using angle bisectors instead of perpendicular bisectors of sides. Perpendicular bisectors give the circumcentre; angle bisectors give the incentre. Position Misconception: Assuming circumcentres always lie inside the triangle. For obtuse-angled triangles, the circumcentre lies outside, and for right-angled triangles, it lies at the midpoint of the hypotenuse.

Circles and Geometric Shapes | Exercise 5.1

Question 1

Draw $\Delta\text{ABC}$ with $\text{AB} = 5\text{ cm}$ , $\angle\text{A} = 70^\circ$ and $\angle\text{B} = 60^\circ$ . Draw the circumcircle of $\Delta\text{ABC}$ . Is the centre inside or outside the triangle?

Solution

Understand the Question

To construct the triangle, draw base $\text{AB} = 5\text{ cm}$ and construct angles $\angle\text{A} = 70^\circ$ and $\angle\text{B} = 60^\circ$ to locate vertex $\text{C}$ .
The circumcentre of a triangle is the point of intersection of the perpendicular bisectors of its sides.
The circumcircle is drawn with the circumcentre as the centre and the distance to any vertex as the radius.
The position of the circumcentre depends on the angles of the triangle:
- Acute triangle: Circumcentre lies inside.
- Right triangle: Circumcentre lies on the hypotenuse (midpoint).
- Obtuse triangle: Circumcentre lies outside.

Step 1 · Construct Triangle ABC

Draw a line segment $\text{AB} = 5\text{ cm}$ .
At point $\text{A}$ , construct $\angle\text{CAB} = 70^\circ$ .
At point $\text{B}$ , construct $\angle\text{CBA} = 60^\circ$ .
Let the two rays intersect at point $\text{C}$ to form $\Delta\text{ABC}$ .

Step 2 · Find Circumcentre O

Draw the perpendicular bisectors of sides $\text{AB}$ , $\text{BC}$ , and $\text{AC}$ .
Mark their point of intersection as $\text{O}$ . Point $\text{O}$ is the circumcentre of $\Delta\text{ABC}$ .

Step 3 · Draw Circumcircle

With $\text{O}$ as centre and radius equal to $\text{OA}$ (or $\text{OB}$ , $\text{OC}$ ), draw a circle.
The circle passes through all three vertices $\text{A}$ , $\text{B}$ , and $\text{C}$ , forming the circumcircle.

Step 4 · Determine Circumcentre Position

Using the angle sum property of a triangle:

\begin{aligned} \angle\text{C} &= 180^\circ - (\angle\text{A} + \angle\text{B}) \\[0.6em] &= 180^\circ - (70^\circ + 60^\circ) \\[0.6em] &= 180^\circ - 130^\circ \\[0.6em] &= 50^\circ \end{aligned}

Since all three angles ( $\angle\text{A} = 70^\circ$ , $\angle\text{B} = 60^\circ$ , and $\angle\text{C} = 50^\circ$ ) are acute ( $< 90^\circ$ ), $\Delta\text{ABC}$ is an acute-angled triangle.

Therefore, the circumcentre lies inside the triangle.

Answer

The centre is inside the triangle.

Common Mistakes

Circumcentre vs Incentre: Using angle bisectors instead of perpendicular bisectors of sides. Perpendicular bisectors give the circumcentre; angle bisectors give the incentre.
Position Misconception: Assuming circumcentres always lie inside the triangle. For obtuse-angled triangles, the circumcentre lies outside, and for right-angled triangles, it lies at the midpoint of the hypotenuse.

Question 1

More questions in Exercise 5.1